A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems

Nopparat Wairojjana,Habib Ur Rehman,Manuel De La Sen,Nuttapol Pakkaranang

doi:10.3390/axioms9030101

Abstract

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones.

Highlights

For a nonempty, closed and convex subset K of a real Hilbert space E and f : E × E → R is a bifunction with f ( p1, p1 ) = 0, for each p1 ∈ K
Many methods have been developed over the last few years to numerically solve the equilibrium problems in both finite and infinite dimensional Hilbert spaces, i.e., the extragradient algorithms [6,7,8,9,10,11,12,13,14] subgradient algorithms [15,16,17,18,19,20,21] inertial methods [22,23,24,25], and others in [26,27,28,29,30,31,32,33,34]
The computational results present this section to prove the effectiveness of Algorithm 1 when compared to Algorithm 3.1 in [39] and Algorithm 1 in [38]

Summary

Introduction

In order to overcome this situation, Hieu et al [12] introduced an extension of the method in [37] to solve the problems of equilibrium in the following manner: let [t]+ := max{t, 0} and choose u0 ∈ K, μ ∈ (0, 1) with ξ 0 > 0, such that vn = arg min{ξ n f (un , y) + 21 kun − yk : y ∈ K}, un+1 = arg min{ξ n f (vn , y) + 12 kun − yk : y ∈ K}, where the stepsize sequence {ξ n } is updated in the following way: μ(kun − vn k2 + kun+1 − vn k2 ).

Preliminaries

Applications to Solve Fixed Point Problems

Application to Solve Variational Inequality Problems

Numerical Experiments

Conclusions